Classical Quantum Mechanics

Abstract

An analysis of quantum interference and Schroedinger's equation
from a classical point of view suggests that
an alternative topology to Minkowski space using hidden dimensions
will allow a single relatively simple wave equation to
have solutions representing multiple particles with all the features of quantum
statistics and spin, as well as wave function collapse, but without the
complexity of quantum field theory.

Carl Brannen,
April 10, 2003

[0] Introduction

There are two paths that physics have achieved success in understanding the world. The first, and most
common, is in the ability to predict the results of experiments. This branch has had successes in the
last century that would have, and did awe the physicists at the beginning of that period. But the
other path is in better understanding the nature of physical reality, and in this, physics has not only
failed to make much progress, but many of the facts that were known in 1903 have since been shown to be
false. The purpose of this short note is to speculate on paths to further this second success. That
is, my purpose here is to suggest ways that we might improve our understanding of the fundamental rules
of reality, rather than continue to improve our calculations for the (admittedly fascinating) details.

Philosophically, the highest success of physics in explaining the fundamental nature of physical
reality has to be relativity, which showed that space and time are part of the same object, and gave
the rules for converting between them. The incredible successes of quantum mechanics in calculating
everything from particle lifetimes to cross sections has not been associated with an equivalent
improvement in our understanding of fundamental reality. The reason for this is fairly clear; after
most of a century, the interpretations of quantum mechanics are not only still unintuitive, they remain
subject to debate. While the nature of the reality behind the experiments is still not just cloudy,
but completely opaque, the calculations for those experiments have been markedly accurate. Quantum
mechanics has been successful at everything except at being understood.

[1] Some difficulties in understanding reality.

Humans have lousy physical intuition.

These difficulties are well known to physicists (who are the intended audience of this note). I'm
including them so that I can give my particular spin on the interpretation of these conundrums, as well
as to bring them to mind.

[1.a] Wave / particle duality.

It's clear that there is both a wave and a particle nature to the world. Relativity works only with
particles, quantum mechanics deals with waves or fields.

[1.b] Proper time vs bizarre time.

Relativity deals with proper time, quantum mechanics can be made to be Lorentz invariant, but never
uses proper time. Instead, quantum mechanics makes a mockery out of time, with particles apparently
travelling backwards in time. The usual interpretation of the two slit experiment is that the electron
does not have a particular location at a given time. This is in contradiction to either the particle
nature of the electron, or the usual understanding of time.

[1.c] Nonlinearity vs linearty.

The mathematical formalism of quantum mechanics, eigenvectors and operators, is linear, but quantum
mechanics itself is nonlinear. The nonlinearity is expressed in the associated rules, particularly
Fermi / Bose statistics, and wave function collapse rules.

[1.d] Causality vs entanglement.

While relativity restricts influences to the speed of light, quantum mechanics entangles particles over
space-like separations. While this is, on the face of it, a violation of the principle that the speed
of light is the fastest influence, relativity is fighting a losing battle by claiming that no data can
be transferred using entanglement. This may be true (good people are working on it), but as far as the
question of the nature of the universe goes, it is clear that causality, as it is usually understood,
is at best, "history".

[1.e] Random vs predictable.

The usual interpretation of quantum mechanics has the results of measurements generally random. But
every now and then the universe provides us with an example where there are correlations that suggest
that it isn't quite so simple. The existence of entangled states suggests that what quantum mechanics
attributes to "probability", could more likely be attributed to "human inability to unravel complicated
entanglement situations".

[1.f] Wave mechanics vs operator formalism.

If you want to make calculations, the formalism of quantum mechanics is the way to go, but wave
equations sure seem like a more realistic description of the universe. The least formal QM equation is
Schroedinger's wave equation. More useful theories require, at the very least, spinors, but provide no
"ether" for those spinors to operate on. Instead, you end up with a wave function where each point in
space has associated with it a collection of (complex) numbers. This just doesn't compare to the
beauty of general relativity and its geometric objects.

[2] Bohmian mechanics.

At night, the hidden variables that are hiding under lamp posts are the easiest ones to find.

[2.a] Introduction to Bohmian Mechanics.

Probably the most popular alternative to the Copenhagen interpretation is "bohmian mechanics". When I
was a graduate student I had never heard of it, but I understand that it's getting more attention
nowadays. But for those who are unaware of it, I'm including a short description here. If you want to
learn more about it from a more reputable source, do a search using google for "bohmian quantum
mechanics" and you will find plenty of hits. If you are not familiar with Bohmian mechanics, here
are some web links that will explain the theory in more detail:
http://plato.stanford.edu/entries/qm-bohm/
http://xxx.lanl.gov/abs/quant-ph/9504010

Bohmian mechanics is an alternative interpretation of quantum mechanics where additional variables are
added. The added variables indicate the actual positions of the particles. That is, the particles are
assumed to always have precise locations. In order to account for interferences, an additional
potential ("quantum potential") is added to the usual (classic) potential. Thus a complete description
of a particle is its wave function and it's initial position. Note that the wave function does not
depend on the initial position, but the particle track depends on both the initial position and the
wave function. This is the most (only?) successful hidden variables theory.

Surprisingly, if you assume that the input states to an experiment follow the usual quantum mechanical
distribution, Bohmian mechanics shows that the output states will also follow those distributions.
That is, under the assumption that the universe is already follows the usual quantum mechanical
distributions, Bohmian mechanics has predictions for the evolution of those distributions that are
identical to those of quantum mechanics. In short, Bohmian mechanics reduces quantum mechanics into a
branch of statistical mechanics. Maybe it would be the standard theory if it didn't have a few
problems.

[2.b] Problems with Bohmian mechanics.

The most difficult to swallow part of Bohmian mechanics is that it is highly non local. The particle
is influenced by conditions in regions which it never approached. But since the predictions match that
of quantum mechanics, the Bohmian supporters argue that this is not a problem, but is instead a
physical necessity. Nevertheless, I find it rather difficult to believe that the universe is set up so
that particles know about regions that are far from where they exist.

Bohmian Mechanics' most difficult problem is said to be that it does not yet have a field theory. A
recent paper by S. Goldstein, D. Dürr, R. Tumulka, and N. Zanghì "Bohmian Mechanics and Quantum Field Theory"
gives a simple field theory for Bohmian mechanics, but it has the same disadvantage that
regular quantum mechanics has, the particle interactions are handled
probablistically. Sheldon Goldstein's articles are linked from his web site here:
http://www.math.rutgers.edu/~oldstein/
But if Bohmian mechanics itself is any guide, a deeper understanding of particle
interactions (maybe in the form of a wave function) will provide an opportunity to replace these
probablities with calculated results from initial conditions. So from the point of view of
understanding the physical nature of the world, I don't find the lack of a field theory to be decisive,
per se. What is more important is that Bohmian mechanics is not relativistic (but see paper cited
above).

A minor problem with Bohmian mechanics is that it implies that the particle trajectories that it
computes turn out to be stationary for some bound states. In particular, the "s" states for a central
force are purely real, and so have no probability flow. These states (and others) therefore have no
electron movement. Of course this is in a playpen version of the theory, and one presumes that an
analysis that included a dynamic vacuum would add enough randomness to walk the electron around the
nucleus.

And I have a philosophical problem with Bohmian mechanics in that it seems odd that the universe would
associate such diverse things as an exact particle position with a wave. For this I have a suggestion,
a way of combining these two disparate notions. I'll discuss it in my notes on Schroedinger's
equation, but the basic idea is to extend Schroedinger's equation to a single (simple) equation that
includes both particle and wave solutions.

[3] Guidelines for fundamental fields.

Symmetry is not necessarily simple.

[3.a] Symmetries can be deceiving.

Since the discovery of broken symmetries in quantum field theory, it's become clear that the symmetry
that nature possesses is not necessarily apparent in what is observed in the physical world. But the
situation is worse than that. There are plenty of examples of very unsymmetrical objects which appear
to have symmetries that they do not, in fact possess. Even crystals, with their explicit breaking of
continuous translational and rotational symmetry, can exhibit (at low energies), isotropic
characteristics.

Symmetries are very attractive to the human brain, and there may be a tendency to exaggerate their
importance. Certainly the periodic table of the elements has many beautiful symmetries, but it would
be difficult to derive Schroedinger's wave equation from those symmetries. Instead, one would have to
happen upon the correct wave equation, and then notice that its solutions possessed the symmetry one
observed.

Symmetries that are apparent at our usual energies may be accidents that obfuscate the underlying
simplicity. In a contest between perfect simplicity and perfect symmetry, the place to put your bet is
on simplicity. For computing matrix elements, imperfect symmetries are great, but the fact that they
are not perfect is an indication that they are probably coincidences, rather than indications of the
fundamental reality.

Historically, most of the advances in physics have come from rejecting symmetries that were previously
assumed clear. The earth was once considered the center of the universe, an obvious symmetry. The
planets were assumed to go around the sun in circular (S1) orbits, or in orbits
defined by the meshing of spheres. Time and distance measurements were assumed to be everywhere equal.
If a Lagrangian possessed a certain symmetry, then it was assumed that its vacuum states would too.
Etc.

[3.b] The mathematics and its interpretation should be very simple, but non linear.

Most of man's successes in physics can be attributed to the success of linear models. This should not
be a surprise; the drunk man tends to look for his wallet under the street lamp first. He does this
because he'll find it quickly, if it's there. And naturally it is the linear approximations that have
had the most success in physics. In addition, those interpretations will likely appear very strange,
as otherwise they would have already been made.

The simplicity of the mathematics will imply that it may not be very useful as far as
predicting the results of high energy particle experiments. This would be consistent with
our present advances in physics. For example, while QED has been very successful at
improving calculations for high energy particles, it has not helped improve calculations
in more prosaic physics such as band gaps in germanium semiconductors. A new dynamics for
extremely high energy situations needs to provide a framework where lower energy dynamics
are explained, but it need not improve the calculations for those lower energy situations.

When relativity first appeared nearly 100 years ago, it did not improve calculations for
much of physics. It was only with the passage of time that it worked its way into the
wide range of physics where it is visible today. We can't expect a new dynamics theory
to be a super theory that explains everything, but instead we should expect it to be
a simple tool that explains a few of our outstanding mysteries.

[3.c] Only real functions should be included.

Any equation that uses complex (or more complicated) numbers can be broken down into real equations.
Complex numbers themselves are no more than a particular symmetry, so their inclusion should not be
seen as a simplifying assumption any more than SU(6) is simple (g). General relativity, for
example, is able to describe the universe rather accurately without the use of complex numbers, and it
is this theory that has given us the most significant peek into the nature of the real world.

I should mention that there is a way of replacing the complex numbers from Schroedinger's equation in a
way that produces a simpler equation. The idea is to replace factors of i in the equation with
d/ds, where d/ds is a derivative in an extra dimension, and simultaneously add a factor
exp(i s) to the wave function. The resulting equation supports all the usual solutions of
Schroedinger's equation, but also includes extra solutions which, depending on various choices,
correspond to particles with higher mass or coupling constant. I'll include more details further on in
this note.

By banning complex functions, I don't mean to exclude them where they simplify the mathematics, only to
distrust them as far as their being fundamental objects. For example, physicists commonly use
exp(it) for a sinusoidal function, taking real parts at the end of the calculation.

[3.d] It must avoid formalism.

Rather than representations of things, we need to have the thing itself. As an example of this,
consider a hydrogen atom. From the point of view of understanding the spectra, the raising and
lowering operators are very useful, but from the point of view of understanding the fundamental nature
of the universe, Schroedinger's equation, with it's explicit use of space and time is closer to
reality.

One of the problems with formalism is that they almost always assume the existence of symmetries. By
using formal theories, one ends up being unable to produce a theory which violates those symmetries,
even if the violation would be below the level detectable with current experiments.

Noether's theorem, which associates symmetries with conserved quantities, is a good example of the
attractiveness of symmetries. The Heisenberg uncertainty relationship suggests that energy is only
conserved on a probabilistic basis. A theory which failed to conserve energy (or momentum), in some
small amount, would therefore not only be consistent with observational evidence, but the existence of
that uncertainty relationship is almost equivalent to a blinking neon sign: "look over here, no one
else did." The universe is limited in terms of its age and size, so Heisenberg's uncertainty
principle places limits on how accurate it is even theoretically possible to design an experiment
intended to test for conservation of energy and or momentum. In other words, not only are energy and
momentum not known to be exactly conserved, present theory holds that they are not, in fact, conserved
exactly at all. Under this situation, would it not be natural that a fundamental theory would have
conservation of these things as an approximate symmetry rather than an exact one? Note: I don't
propose to repeal these conservation laws, I am making this point only to give the professional reader
a reason to briefly consider theories not written in the formalism of Lagrange or Hamilton, and perhaps
not writable at all that way.

[4] Hypotheses for a fundamental theory of mechanics.

What IS the sound of one hand clapping, and why are we teaching this to our students?

[4.a] Begin with Bohmian mechanics.

Bohmian mechanics has the advantage of having a single interpretation. The more formal theories have
the disadvantage of being too tied into symmetries that at the very least may hide an underlying
simplicity, and at the worst may not be exact.

[4.b] Use hidden dimensions when they simplify the model.

By hidden dimensions, I mean stuff along the line of Kaluza-Klein, where relativity plus an added
dimension gives electricity and magnetism. String theory has also had some success by bringing in
hidden dimensions, but string theory has the same weaknesses as the formalism that gave birth to it.
None of the usual quantum mechanical paradoxes is resovled by string theory, except for UV issues.

[4.c] Let the particle go through both slits.

If the underlying mathematics is going to be simple, it's going to have to be interpreted in only one
way. In standard Bohmian mechanics, the particle is treated as a point particle with exact positions
at any given time. Under this circumstance, it doesn't make much sense that the particle could be
influenced by goings on many lightyears away, but if the particle position is treated as the result of
the wave function collapse, the theory becomes more intuitive. That is, the paths that the particle
can make are influenced by interference, but after the particle makes its choice (which I am guessing
has something to do with the proper time experienced by the particle), its path resolves to a single
choice. Maybe a way of putting it would be to say that the particle "thinks" about going through each
slit.

This can only be accomplished if our usual understanding of either particles or time is modified.
Before the particle actually makes its voyage its wave function is aware of all the choices, but as
time goes on, the wave function collapses on to the path actually chosen. The current standard
interpretation of Bohmian mechanics makes this possible by by defining the particle to have two parts,
a wave function and a position. This is more complicated than I'd like. I think that there is another
way of accomplishing the same thing, and that is to consider the position of the particle as defined by
a wave collapse that occurs as proper time transpires. This leaves the particle able to be influenced
over space-like separated regions before the wave is collapsed, and as time passes, the particle's wave
is collapsed down to its position. Long after the experiment has completed, we can observe the
particle's track, but we can also see the influences of the wave nature of the particle. The result is
that the particle's wave did traverse both slits, but as time progressed, the particle's path became
frozen and a particular path was selected.

[4.d] Use proper time in quantum mechanics.

If it is the passage of time that causes a wave function to collapse, then by the principles of special
relativity, that time will have to be proper time. But neither quantum mechanics nor relativity
include proper time as a dimensional variable. Instead, proper time is a derived quantity. If proper
time is how wave functions collapse, then it needs to be brought into quantum mechanics in a more
direct way. Under this interpretation (i.e. Bohmian mechanics with proper time as the wave function
collapse mechanism), the standard interpretation of quantum mechanics looks only at the physical
situation at an "advanced time" (to misuse the concept of "advanced" somewhat), that is, before the
particle traverses the slits, while the particle track exists in "retarded time".

Adding a decay rate that is proportional to proper time does add complexity, but it may be that there
is a simple underlying equation that can be interpreted to have a standard quantum mechanical wave part
for the time before the particle goes through, and also a decay to a collapsed wave function
afterwards. Certainly it's no worse than having things as they now stand, with waves and particles
having to share the fundamental spotlight.

Note that proper time is associated with individual particles. It is not an attribute of a reference
frame or a collection of particles, as each of those particles may have different ages.

[5] Some notes on Schroedinger's equation

Who taught Mother Nature to do complex arithmetic?

For reference, Schroedinger's wave equation for a scalar particle in a potential V(r):

(5.1) i h ∂Ψ / ∂ t =
(- h / 2m Δ + V(r)) Ψ
(r,t) .

Before going into further discussion, I should first note that this equation applies only to
experiments that have not yet been run. That is, it is a postulate of quantum mechanics that after you
run an experiment, the wave function is collapsed into whatever you end up observing. In that sense,
the t variable in the above equation can only be interpreted as applying to a time in the
future, not the past.

Over the years more than one mathematician (including my father who bought the copy of Messiah quoted
later on) has asked me more than once why it is that the wave function is complex and requires an
absolute value and squaring operation before it can be interpreted as a probability. I think this is a
useful question to address.

Before a student can understand (standard) quantum mechanics, he must first study classical mechanics.
Classical mechanics originated with analyzing a collection of masses, with each mass having a position
and velocity (or momentum). These are governed by Newton's three laws, which are still taught.
In order to more simply analyze situations where particles are constrained (by ropes, for example),
and/or where it is desired to use more arbitrary coordinates (spherical coordinates for central forces
problems, for example),
Lagrange's equations of motion are used. In these 2nd order equations, the positions and velocities
become "generalized coordinates" and "generalized velocities" and the constraint equations are no
longer apparent but are implicit. The Lagrangian equations of motion can be transformed to Hamiltonian
equations, which replace the 2nd order differential equations with twice as many 1st order equations.
It is perhaps indicative of a certain mindset in theoretical physics, that the variables now containing
the information about positions and velocities are known in the Hamiltonian formalism as "canonical
coordinates".

The Lagrangian or Hamiltonian formalism can be naturally extended to continuous systems
of matter. It is this continuous extension that is directly connected to quantum
mechanics.

The Hamiltonian formalism for continuous systems requires two functions to describe the field. That
there are two functions is why the quantum wave state cannot be described with a single real valued
function (though I will later in this chapter show an extension of Schroedinger's equation that does
require only a single function). Having a single real
valued function could provide a probabilty density, but it would not also be able to provide a velocity
at the same time. Yes, I know that this more or less directly contradicts my previous sentence, but to
get a velocity out of a real valued function I have to add an extra dimension.

Note that the Hamiltonian coordinates are generalized coordinates. This means that there is no
particular reason why they should correspond to a probability density as opposed to a complex square
root of a probability density. In fact, it's easy enough to rewrite Schroedinger's wave equation so
that one of the two wave functions is the probability density (as we will see in the next section).
These observations suggest that the question of "why does the quantum wave function require squaring a
complex absolute value to get a probability density" could instead be written: "why is the complex
version of the quantum wave function so simple?".

One of the most fruitful ideas of physics has been the principle of linear combinations. Sine waves
are particularly useful in physics because they can be added together in linear combinations to create
more sine (or cosine) waves. Another very fruitful assumption in physics has been the principle of
perturbations. The combination of these fruitful assumptions suggests that a simple theory of quantum
mechanics would arise from equations which (a) support sine waves, and (b) are linear. But sine waves
go negative, so they can't be used as probability densities. The simplest way to convert such a wave
to a legal probability density is to square it.

Given the above, and recognizing the formal nature of quantum theory, it is natural that the quantum
wave state be given by a complex valued function. But one might also ask if this is a case of the
missing wallet being searched for under the lamppost. The nonlinear, non perturbed parts of physics
are the more complicated ones, so it is natural that our dearest successes have been in the areas where
linear combinations and perturbation analysis work best. This is not in the region of time where the
wave function has collapsed and is concentrated, but instead in the future, where the wave function is
as dilute as possible.

There is another clue from the above notes, and that is that the formalism of quantum mechanics is
based on principles of classical mechanics that involve constraints. This suggests that when we look
for more fundamental descriptions of quantum mechanics that involve extra dimensions, we look around
for more or less natural constraints that can be added at the same time. In particular, what kind of
constraint would turn a simple version of Schroedinger's equation (using the probability density and
velocity field version) into a simple but less linear equation perhaps using more dimensions?

[5.a] Schroedinger's wave equation as an attractor

Since Schroedinger's wave equation (for scalar particles) is the most simple quantum mechanical wave
equation, it is a good idea to understand it thoroughly before trying to replace, say, QED, with
something more fundamental. The natural inclination is to convert Schroedinger's complex equation into
two equations in the reals. There are two obvious ways that this can be done. We can write
Ψ = x + iy, or we can write Ψ = Rexp(i
θ).
The first way leads to two coupled equations, similar except for a minus sign or two, and no obvious
way of interpreting it physically except as two coupled equations. But splitting
Schroedinger's wave equation into x + iy and adding a hidden dimension to
allow it to be written as a single real equation does lead to some interesting insights, so
I've provided some notes on this subject in Appendix B. The second separation,
Ψ = Rexp(i θ), is the more fruitful and it is that
separation that I will pursue for most of this paper.

Rather than convert all the gory details into HTML, let me quote Messiah who does pretty much
what we want, but uses
Ψ = Aexp(i S/h )
and then goes on to take the classical approximation h -> 0. My copy is apparently the
first English edition (John Wiley ~1961), look in pages 222-224 of volume 1, in the beginning of the
section entitled "Classical Limit of the Schroedinger's Equation".
Making this substitution, and seperating into real and imaginary parts, and multiplying
by 2A gives two equations, one for ∂S/∂t
[refer (VI.17) in Messiah], the other for ∂A/∂t [refer (VI.19)]:

(5.2a) ∂S/∂t + grad2 (S)
/(2 m) + V
= (h2 / 2m) (Δ A) / A,

(5.2b) m ∂(A2)/∂t
+ div(A2gradS) = 0.

Equation (5.2b) is simply the continuity equation for the probablity density A2. It
can be interpreted in a purely classical manner. Messiah interprets S to be a
potential that generates a velocity field v for that probability density (where
J is the usual probability current density) [refer (VI.21)]:

(5.3) v = J/A2 = grad(S) / m.

Messiah then shows that (with h = 0) the velocity field follows the law of motion for a
classical fluid under the influence of the given potential. But if he had not taken the classical
limit, the conclusion that S was a potential for a velocity field would have still been valid,
as there is no h in the probability continuity equation (5.2b).

Since the probability density acts like a classical fluid, as opposed to a gas, it need have only a
single velocity defined at each point. This is not what a statistical ensemble of (non interacting)
particles with various positions and velocities would give.
Such a collection would be a gas, rather than a fluid.
Instead, the "particles" strongly interact, and they interact in such a way as to allow only a single
velocity at each point in space.

S only shows up with a gradient, except for the term indicating how it changes with time. It is
therefore possible to take the gradient of that equation (5.2a), and then make a change of variables to
replace grad S with m v:

(5.4) m ∂v/∂t
+ grad(v2 / 2m) + grad V
= (h2 / 2m) grad(Δ (A) / A).

As an aside, you can work the right hand side of the above equation to use
A2 instead of A, but the result is about as complicated, and the
final result can be written in several distinct ways. Also note that the only term that is not already
a gradient is the ∂v/∂t term. This means that the above equation can be
used to compute the time evolution of velocity fields which are not the gradient of any S.

Equation (5.4) is useful in that Schroedinger's equation has automatically built into it the assumption
that Δ X Δ P ≥ h. So if you wanted to see what the time
evolution of a wave state that was initially more tightly concentrated than Heisenberg's uncertainty
principle would allow, you cannot evolve the function using Schroedinger's wave equation. That is, if
you minimize ΔX by packing all of A2's support into a very small region,
you end up with ∂/∂x being very large.
But by using the above equation, you can initialize both the position and the velocities to whatever
you like (and use v explicitly for the probability density velocity). Of course if your
initial conditions satisfy Schroedinger's wave equation, then the computed evolution will match
Schroedinger's. Since all the changes to mv can be written as a
gradient, one would expect that the discrepancy between mv and the velocity field
that has a potential, and therefore that can be associated with Schroedinger's wave equation
would decrease with time as the effects of the initial conditions recede into the past, though
I have not derived this.

Equation (5.4) also suggests a way of combining the particle and wave in Bohmian mechanics, and thereby
make more explicit the probability wave collapse. As I mentioned earlier, Schroedinger's wave equation
is suitable only for experiments that are in the future. But equation (5.4), which includes
Schroedinger's, also has solutions for waves with position and momentum both as accurately described as
needed. This makes a wave function that is more compatible with our own intuitive nature of time.
Only the future is nebulous, the past cannot be changed. But while we gloss over the transition
between past and future, equation (5.4) allows that transition to be made explicit.

In this interpretation, the particle's position and momentum were perfectly (at least exponentially
perfectly) defined in the far past. The present corresponds to a period where the error in position
and momentum exponentially increase, until they knit into the future where Heisenberg's uncertainty
principle applies completely. Equation (5.4) supports solutions to the wave equation with this
property.

But there are problems with equation (5.4), chief among them being that it doesn't have the right
statistics or spin. That is, when you translate the interference that we all know that quantum waves
experience back into equation (5.4), you end up with a horribly complicated mess. So while (5.4) gives
the feel for what I believe we should be searching for, it is not a very direct signpost to that spot.
On the other hand, equation (5.4) is compatible with the usual quantum mechanical spin, and it supports
solutions with SU(2) symmetry without the need for a complicated or arbitrary topology for space-time,
or for functions with more than the usual classical complexity. More on this subject in a later
section, where we will look at the classical interpretations of angular momentum and spin.

I should note that there are undoubtedly a lot of other equations that have Schroedinger's wave
equation as an attractor. Maybe one of these is such a simple equation that it is a more natural
candidate for a fundamental theory. But given the problems with a non relativistic spinless theory,
I'm not inclined to spend a lot of time searching for one.

[5.b]Interference in Schroedinger's wave equation

Interference between alternate pathways is one of the primary features of quantum mechanics, and since
Schroedinger's wave equation has a classical limit, it makes sense that we should take a look at
quantum interference for Schroedinger's.

The first thing to note is that Ψ is a complex function, so its interference shows up in the
form of a complex addition. The addition, when expressed in terms of our real variables A and
S, is rather complicated. Why does it work so well? We all know where freshmen in college
learn about adding complex numbers, but I fail to see who taught mother nature the details. The
explanation that "mathematics models reality amazingly well" is true enough, but I'm looking for an
understanding of physical reality, not a good model.

My guess is that Schroedinger's wave equation is a formalism rather than a fundamental part of nature.
That is, the complex numbers are a symmetry that is not necessarily obvious in the underlying
fundamental reality. More fundamental theories, like relativity, get by just fine with the reals,
quantum mechanics should too. (Try finding a complex number in Misner Thorne and Wheeler's
Gravitation.) Unfortunately, with quantum mechanics, Schroedinger's wave equation is the most
fundamental object I've got, so any analysis has to start here.

Rather than looking at the 2-slit experiment, I think we will make better progress by looking instead
at interference effects between two particles. The interference in the 2-slit experiment is
understandable strictly as a wave effect, and as such isn't any more interesting than the similar
effects that show up with light. I've already concluded that the fundamental nature of matter is
waves, so the 2-slit experiment doesn't surprise.

But the interference effects between two (or more) identical or non identical particles are more
interesting. There are four cases worth running through the calculations on, identical fermions (two
electrons), identical bosons (two pions), distinguishable particles (an electron and a proton), and
identical classical particles. Note that all this talk about indistinguishable particles is perfectly
consistent with our thinking of them as some sort of waves. Of course waves are indistinguishable. If
we instead thought of particles as point particles I would think that they would still be
indistinguishable, but by assuming that the fundamental nature of them is a wave this doesn't come up.
Waves is waves.

The quantum mechanical fundamental description for a single particle consists of its wave function,
whose informational content includes a probability density and a velocity potential. The corresponding
description for two particles is a joint probability density, and a joint velocity potential. The
joint probability density is a function on R3 x R3, as it has to
provide a density value for the two particles. The two position coordinates will be called
x1 and x2. To be specific, let P(x1,
x2) d3x1d3x2
will be the probability that the the first particle is found within
d3x1 of the position x1, and the second particle
within d3x2 of the position x2.

I've already made the more sensitive wince, by refering to "first" and "second" particles just inches
from having previously assumed that they are identical. Because of this defacto enumeration of the
particles, most of our calculated probability density values will be 1/2 (for n particles it
would be 1/n! ) the literal value, (but they'll be correct when x1 =
x2). The reason physicists use this way of counting instead is that it makes the
wave functions easier to normalize and calculate with. That is, if our space only had two points,
a and b, then we will have P(a,a) + P(a,b) +
P(b,a) + P(b,b) = 1.

We'll compare the four cases using notation common to the quantum case. So let
P1(x) and P2(x) be the the probability densities
(i.e. P = A2), and S1(x) and
S2(x) be the velocity potentials for the corresponding quantum particles.

If the particles are distinguishable, the combined probability density is just the
product of the densities:

(5.5) Pdist(x1, x2) =
P1(x1) P2(x2)

If the particles are identical, but are classical, we have pretty much the same thing, but since the
particles are identical, we have to take this fact into account. Note that the factor of 2 is from our
splitting the density for x1 ≠ x2 into to two parts; the factor
which may have caused wincing:

(5.6) Pclass(x1, x2) =
(P1(x1) P2(x2) +
P1(x2) P2(x1)) / 2.

The formulas for fermions and bosons are more complicated as they also depend on A. In these
equations, the ± indicates bosons (+) or fermions (-):

The complexity of the interference term is consistent with
quantum mechanics being a perturbational
theory. That is, the theory is beautifully linear as long
as no particles risk being put into the same
position, but the interaction between two particles is
highly non linear.

Note that S1 or S2 could have a constant added to it
with no change in the result. This is a global symmetry of S. But there is also
a local symmetry in that local changes to S by multiples of 2πh
result in no
change in the joint probability density. These observations also apply to the joint
velocity potential, Squant(x1, x2).
Also note that if P1 and P2 share no support, then
the quantum mechanical formula is identical to the classical one, which supports the decision to divide
Ψ into two real functions.

The fact that the joint wave function does not depend, even locally, on changes in S by
2πh
suggests that S should not be considered to be a real function. Instead, it's natural range
should be S1. This explains, in a circular
manner, why it is that mapping of P and S into complex form results
in such a simple wave equation.

Also note that if one of the two wave functions has a constant added to S, there is no change to
the resulting interference effects. That is, S1 only shows up as the difference in
its value between two points. A global addition of a constant has no physical effect. This fact
suggests that there is a simpler way of looking at spin than to represent it as an SU(2)
symmetry.

From the discussion so far, one could get the impression that S can be a well
defined real function of real space, but a brief consideration of bound states for the hydrogen atom
with non zero angular momentum show that this is not the case. The simplest such state is the p-wave
eigenfunction with angular momentum in the z direction
of 1. Since life is short and HTML is long, I'm leaving off the normalization factors,
and using spherical coordinates:

(5.8) Ψ211 = rexp(-r/2a0)
sin(θ) exp(i φ).

Note that the above equation ignores the dependency on t, which I'll include back
in later. For now it's sufficient to ignore time dependence, and to look at the
interference effects at only a single moment in time.
From equation (5.8), it's clear that S211 depends only on φ, and that it depends
on φ
in a discontinuous manner. That is, S is discontinuous at φ = 0 == 2π. This fact
suggests that the natural way to implement Schroedinger's wave equation is with the v
variable instead, using equation (5.4).

Since v is defined by the gradient of S, converting equation (5.4)
into an equation for v instead of S requires an integral. Since the
integral is a path integral, it's not obvious that the result does not depend on the path
chosen. Of course if our solution is also a solution of Schroedinger's equation, then
the existence of the velocity potential S implies that there is no path dependency.
If we restrict ourselves to cases where x1 and x2 lie on the same
trajectory, we can use the trajectory to define S from v without having this
worry. But since Schroedinger's
has this fact (that an S exists for any "physically possible" v)
built into its equation, I'll ignore this problem for now. The resulting equation
for quantum interference, as defined by velocity instead of velocity potential, is
therefore as follows (with a very small amount of arithmetic), and I've chosen to
parameterize the path integrals by η, with units of distance.
This way the units are made obvious:

That (5.9) seems less complicated than one might expect is a good indication that we're
not getting too far away from the trail of a simple equation with Schroedinger's wave
equation as a limiting case. But about that cosine function. While a cosine is to be
expected in any interference calculation, this interference is not the usual type. To
contrast this interference from the type of interference we're used to in classical
equations, it's useful to examine a classical interference, say between
two cosines simply added together in the usual way. To simplify this a bit, let's take
the two waves to both be cosines, both with unity amplitude. The more general case
with various amplitudes of sines and cosines gives frequency results that are similar,
but this case is sufficient to detail the interference.

(5.10) cos( A ) + cos( B ) = 2 cos((A + B)/2) cos((A - B)/2).

Equation (5.10) has the usual interference terms that we come to expect. Two frequencies
are present in the combination. The first carries the average frequency and corresponds
to a wave with an intermediate wave vector. The second depends on the difference, and is
the inteference term.

Comparing to equation (5.9), we see that quantum interference is similar to classical
interference, but only the A-B term is present. The two cosines on the left hand
side of the equality in (5.10) do not show in (5.9), nor does the A+B term on the
right hand side.
The missing cosines are present only in the complex description of the wave function, they disappear when the
wave function is split into the two classical parts. This is a little confusing, so let me explain further.

A quantum mechanical plane wave such as exp(i(kx-wt))
represents a steady flow of particles in a single direction, with the
particle density everywhere equal. There is no sine or cosine dependence.
Another way of describing this is to note that if one
slit of the two slit experiment is covered up, the resulting distribution
of particles on the screen is even both in space and in time. There is no physical significance to the
oscillations in the quantum wave, except when
the wave interferes with itself or others. Thus from a physical point of view
there are three missing cosines/sines in the equation for quantum
interference, as compared to classical interference.

So while quantum interference appears to be very distinct from
classical interference, the absence of those three cosines
may be due to a limitation of Schroedinger's equation. That is, the missing cosines
may show up in a more fundamental theory, but are missing in Schroedinger's wave equation
because their frequencies are too high. Instead, these very high frequencies may be
averaged over in Schroedinger's equation, so only the low frequency interference
term cos(A + B) appears.

So we go back and look at the interference term in equation (5.9) once more, and look
for an alternative explanation for that interference, one more radical than classical
wave interference. The explanation has to involve a structure that somehow combines
velocities and positions, and it will probably result in either or both of those
being quantized.

What's really odd about this interference is that it is completely cyclic in
h. That is, if you are considering points that are at a distance Δx,
an addition to the momentum mv1 of h /Δx results in
the same inteference. What's also significant is that you can
change the intervening momenta (that is, the momentum in the positions between x1 and
x2), and provided you change it by an amount that integrates out to 2π, there is
no way to detect the difference at the end
points.

Of course it's an accepted fact that angular momentum is quantized. But the cosine
factor in equation (5.9) suggests that there is a milder form of a sort of quantization
for momentum itself. That is, that momentum integrated over a distance interferes with itself
according to multiples of h. That this relation involves both
position and momentum suggests that it is an artifact of a deeper relationship between
those two.

[5.c] Quantum angular momentum and spin

Sometimes things are not as complicated as they seem.

It's frequently said that angular momentum in classical mechanics is analogous
to quantum mechanical angular momentum, but that spin is a purely quantum effect.
Spin adds the complexity of SU(2) symmetry to quantum mechanics, and it does
it in a way that is difficult to extract. So to get to the fundamental physics of spin
it is best to approach it first from the more convenient study of angular momentum.

One of the features of the half integer representations of SU(2) is that they
require rotations by 4π in order to come back to return to the the unrotated situation.
This feature shows up in spinor representations of fermions, and at first glance it
suggests that the world is a very odd place. Classically, rotations by 2π leave
the system unchanged, but with spinors, the wave function is negated.

Classically we can get along with just positions and velocities, can we achieve quantum
spin with just these variables? Or do we have to follow the spinor representation and
end up with a multiply valued fundamental representation of reality?

As was noted in section [5.b], changes to the wave function by multiplication by
a constant do not change the physical situation. They correspond to the addition of a constant to
S, and by equation (5.7), there is no resulting interference
effect. Moreover, replacing Ψ with P and v
completely removes the need for the spinor representations. It's easy enough to create
a wave in P and v that has spin 1/2, in fact it's all too easy
to create waves with perfectly arbitrary spin. Presumably the difference between particles
of various spin is in the details of their wave functions. That is, while the Schroedinger
wave equation enforces integer spins for (large) composite objects, it's possible that
the equation is only a space averaging approximation for an equation that is applicable to
smaller dimensions, and includes half integer spins available only in these smaller sizes.
Later on in this paper, I'll show just such a space.

This all suggests that the R3 that makes up the obvious part of our
real world is not the complete (local) topology of the world,
but that there is at least one
hidden dimension as well. This more complicated topology is going to have to be consistent with
relativity, and support the odd quantum
interference effects as well.
Since we understand the principles of relativity a hell of a lot better than those
of quantum mechanics, the place to start looking for this alternative topology is
in relativity. Eventually it would be nice to include general relativity and
gravitation, but special relativity definitely comes first.

[6] A Topology for Proper Time.

To make an omelette, you will have to start by breaking a few eggs.

As discussed in section [4], the fundamental time of the universe appears to be
the proper time that is
distinct for each individual particle. That is, the global time that we use in our
equations seems to have the character of a convenient parameter to integrate over
rather than a fundamental part of the world. So our modified topology should include
proper time. And as integration is still a useful thing, we'll include global time
as well. This suggests that we begin with the standard Lorentz metric, which allows
the computation of proper time in terms of position and global time:

(6.1) ds2 = dt2 – (dx2 + dy2 +
dz2).

While the abover formula is convenient for computing proper time,
it doesn't show proper time as a independent dimension, so let's imitate
Einstein a bit and instead treat proper time as if it were a
spatial dimension. That is, relativity shows that space and global time are mixed
by boosts. Our topology will instead mix space and proper time, and leave global time
for calculational purposes only. The metric that we'll use is therefore:

(6.2) dt2 = ds2 + dx2 + dy2 +
dz2

The above relation gives the amount of global coordinate time required by a particle to move (dx,
dy, dz) given that it experiences a proper time interval of ds.
It has the advantage over (6.1) that it is positive. So the local topology, instead
of being Minkowski, will instead be Riemannian. This is a simpler topology than that
of (6.1), so we can expect that some of the odd features of the usual topology of
special relativity will vanish from the local topology, and will instead show up only in the global
topology.

With the local topology selected (i.e. equation (6.2) which gives R4),
we must now choose the global topology.
We have proper time as one of the coordinates (or dimensions), but we have some
interpretation issues.
Different particles can experience proper time at different rates, but can nevertheless
end up at the same positions. So the proper time dimension must be one that is ignored
from the point of view of comparing positions. That is, to compare two positions (to
see if there is a collision, for example), we must ignore the proper time coordinate.

Giving the proper time a cyclic coordinate will allow different proper times to
be associated with the same (x, y, z) position. More formally,
choose S1 as the topology of the proper time dimension, so that the
full topology for points in the space such as (x, y, z, s)
is R3xS1, with the S1 circle having
a circumference of L. In this topology, the shortest (global) distance between
two points will be influenced by the hidden dimension only by, at most, L/2, and
so will be neglible for sufficiently small L. What's more, an error in distance
of the maximum amount will occur only if the points occupy the same real coordinates.

The Proper Time topology violates the basic assumption of special relativity that
all frames of reference are equivalent.
That is, the topology defines a frame of reference that is the prefered one. But since
this topology uses the same metric as Minkowski space, the fact that there is a prefered
frame of reference may not be observable, but instead is as hidden as the hidden dimensions
that allow this topology to function. Special relativity is an approximate symmetry
of this topology instead of an exact symmetry, as will be shown later. For those who
have trouble accepting that this topology does transform frames of reference the same
way as special relativity does, nothing can be more convincing than calculations, which
are provided in Appendix A.

This Proper Time topology is perhaps more complicated globally than Minkowski space,
but since the metric has no minus signs it is a standard differentiable manifold, and is
therefore considerably less complicated locally. The standard results of manifold theory
will hold in this space. This is exactly the kind of tradeoff
that we're looking for. That is, since the electron is (according to current
experimental data) a point particle, if we are looking for a version of Schroedinger's
wave equation that gives this particle its h/2 spin, we must
add a complexity to the
local nature of space, but that complexity must disappear for larger distances.

In the Proper Time topology, the magnitude of the velocity of any particle is
c (= 1).

(6.3) |( dx/dt, dy/dt, dz/dt,
ds/dt)| = 1.

Another way of deriving the above relation is to note that if the velocity of a
particle in the 3 real dimensions is β then by time dilation, the ratio of
velocity in the proper time dimension is sqrt(1 - β 2), so the total velocity of
the particle is 1. This relation
is built into the metric chosen for the topology, so it should not be a surprise.
In fact, you could do the same thing with any space with an arbitrary mechanics,
provided the particle speeds have an upper bound. What makes this topology useful for
our own universe is
that proper time is such an important part of special relativity, and
the relations that connect proper time to the regular dimensions make the metric
particularly simple.

Having all particles travel at the same speed (at least locally), is convenient for
calculational purposes, and we can expect that some of the relations of special
relativity (such as conservation laws) will be simplified in this topology (and the
simple derivations will be provided later). Again,
this kind of simplification is what we want, and besides, since it is clear that
matter is carried by waves, what would be more natural than to have all waves travel
at the same speed? Waves, at least conceptually, require an "ether" to propagate in,
and the most natural ether is one where all waves travel at the same speed. Anything
else requires complicated dispersion relationships.

In the proper time topology, the proper time experienced by a particle is measured by
how far it moves in the s dimension. While the s dimension is only a
circle of circumference L, it is possible to go around that circle many times,
so very long proper times can be represented by many orbits around that circle.

The defining metric for the proper time topology (6.2) is identical to the metric
for standard special relativity, and so the two theories give the same answers when
computing times and distances, (at least when ignoring very small discrepancies in
positions), but for those readers that are unsure of the concept, or for those who
want to see how time dilation and length contraction are computed in a proper time
topology, as opposed to the global time topology of special relativity, I've included
sample calculations for time dilation and length contraction in Appendix A.

[6.a] Classical Mechanics in the Proper Time Topology.

Before we get into looking at how waves interact with the proper time topology, we
should take a look at how special relativity looks in it. One expects that the
laws of special relativity will be simplified in this topology, if it is a natural
one. This would also be a good spot to collect together the correspondences between special
relativity and proper time.

Please forgive me for setting c = 1, but then later showing it explicitly, in an
inconsistent manner, as well as my switching between a space-like and a
time-like metric. Proper Time topology eliminates these confusions as the metric is
fully positive, and all dimensions are naturally measured as distance.

Special Relativity vs Proper Time

.

Special Relativity

Proper Time

Points:

(ct, x, y, z)

(x, y, z, s)

Local Topology:

Minkowski

R4

Global Topology:

Minkowski

R3xS1

Local Metric:

-(cdt)2 + (dx)2
+ (dy)2 + (dz)2

(dx)2 + (dy)2
+ (dz)2 + (ds)2

Velocity:

(c, dx/dt, dy/dt, dz/dt)

(dx/dt, dy/dt, dz/dt,
ds/dt)

Energy/Momentum:

m(c, dx/dt,
dy/dt, dz/dt ) / √(1 - β2)

m(dx/dt, dy/dt, dz/dt,
1 ) / √(1 - β2)

Energy/Momentum:

m(cdt/ds, dx/ds,
dy/ds, dz/ds )

m(dx/ds, dy/ds,
dz/ds, dt/ds)

Of course the conservation laws are unchanged. This is not a simplification for
these laws, but the thing to note is that conservation laws are consequences
of symmetries, and symmetries cannot be trusted. So the fact that Proper Time
doesn't simplify the conservation laws is not significant in terms of its being
an acceptable alternative. What's important is that the local topology, the place
where the weird quantum physics takes place, is simplified.

The assumption that all frames of reference are equivalent,
which is the fundamental symmetry law that underpins special relativity, is replaced
in Proper Time
with the assumption of the combination of a special topology and a constant speed for
all particles. Instead of a theory based on a symmetry relation, Proper Time Topology
is instead based on an assumption of simplicity. For example, since all electrons
in Proper Time are travelling at the same speed, (at least locally), it is reasonable
to assume that the waves that represent them have a characteristic frequency, one that
does not depend on the velocity of the electron.

Intuitively, Proper Time has some simple explanations for some of the oddities of
our universe. It's clear why we can't accelerate objects to faster than the speed of
light, as that is the speed they are always travelling at. The huge amount of energy
present in even stationary objects (i.e. E = m c2) is
explained by their velocity in the hidden dimension. The distinction between massive
and massless particles disappears, (except in the conservation laws).

Note that in special relativity, a particle's position, as a function of the global time,
can be defined with just 3 numbers, but in Proper Time a fourth number is required.
This means that when converting from special relativity to Proper Time, there is an
undetermined variable. But since the error in this variable cannot be more than L/2,
the effect of this undetermined variable will be negligible in practice.

[6.b] Self Interference Effects in Proper Time Topology.

Since point particles experience only a local topology, a classical universe consisting
of point particles only will be unable to distinguish between Proper Time and special
relativity, except to the extent that distance measurements are very slightly increased
(i.e. Euclidian geometry will be slightly inaccurate). But waves
in Proper Time will intefere with themselves and with other waves, and this may be
observable in the real world, depending on the size of the S1
circumference L.

Let W(x, y, z, s, t) be a plane wave function that represents a wave moving in the
(kx, ky, kz,
ks) direction. In order for the wave to travel at the velocity c,
we need the frequency ω to be equal to k c, where k is the length of the
k = (kx, ky, kz, ks) vector. A general form for W is
W(k • x – ω t), where W
is some cyclic
function with period 2π. For a linear theory, we might try a cosine function, but in
general the function is going to have to be highly non linear in order to get the right
quantum statistics. I'll leave the interpretation of W for later. For now, suffice
it to note that W is a general plane wave.

In order to satisfy the Proper Time topology, W must be single valued over s.
This implies that Lks = 2nπ for n an integer.
The wave vector is therefore quantized, at least in the proper time direction, according to:

(6.4) ks = 2 n π / L.

This implies that the speed in real space R3, as well as the time
dilation ratio, Rtd =
√(1 – (v/c)2)
= ks / k is also quantized:

This result is very heartening, because the classical nature of Schroedinger's wave
equation (see section 5.b) implies a quantization of velocity, and some sort of hidden
length scale.

There are some other odd consequences of this quantization. The number of different
possible speeds Ns is no longer infinite as in special relativity,
but instead is finite (for a particle with a given characteristic frequency ω):

(6.6) Ns = [ω L / (2 c π)].

Where "[]" denotes greatest integer less than or equal. Of course for this theory to
be realistic, Ns must be very large, so in the remainder of this note,
it is so assumed.

Since there are only a finite number of different speeds, there must be a
maximum speed, and this maximum is attainable, but is less than c:

(6.7) Vmax
= c √(1 - (2cπ/Lω)2)
~= c - 2c(cπ/Lω)2.

There is also a minimum attainable speed, which may be zero, or may be larger, depending on the exact
values of ω, c, and L. But the minimum attainable speed will
be bounded above by:

(6.8) vmin < c √( 4cπ/Lω).

The above equation (6.8) gives the quantization of speeds for speeds near zero.
That is, low speeds are quantized approximately according to:

(6.9) vslow
= nc √( 4cπ/Lω) + vmin.

The quantization in velocity has a consequence in the equations of motion for very
slow moving waves. The continuity relation that enforces the quantization of velocity
has to be applied at a rate of once per each revolution that the particle makes in the
S1 dimension. This implies that the slowest possible acceleration
that a wave can be given is defined by a force that changes the velocity of the wave
by one velocity step (i.e. equation 6.8) per revolution. Since the particle is
assumed to be travelling at close to c in the s dimension, the result
is that this characteristic minimum acceleration is given by:

(6.10) amin
= c2 √( 4cπ/Lω) / L
= √( 4c5π/L3ω).

Since it is impossible to achieve an exact velocity, it is also not possible to define
the kinetic energy exactly. From this we can conclude that the wave/particle travels so
as to either cause its kinetic energy to become more than the potential energy would
indicate, or so that its kinetic energy becomes less. But since there is a minimum
velocity, and this velocity is unlikely to be zero, it would make sense that the particle
would tend towards choosing the larger velocity, and therefore the larger kinetic
energy. The result is that very small forces will drive larger than expected
accelerations. That is, the wave will tend to accelerate more than predicted by special
relativity.

In order for this topology's effects to have gone unnoticed, it's clear that L
must be almost vanishingly small. This requires ω to be very large in order to
avoid the quantization effects of equations (6.4-6.10).

The physics literature does have mention of a minimum acceleration effect known
as "Modified Newtonian Dyanmics" or MOND, but it
appears to be such a large effect that it is not consistent with equation (6.10),
at least when the Proper Time topology is used to explain the interference terms
in quantum statistics.
MOND was developed as an alternative explanation for the rotational curves of galaxies,
which most astronomers ascribe to dark matter. The original paper is Mordehai Milgrom “A
modification of the Newtonian dyanamics as a possible alternative to the hidden
mass hypothesis”, Astrophysics. J. 270:365-370 (1983), and a search of the internet
for the terms "MOND" and "Milgrom" will provide plenty of hits. The characteristic
acceleration in MOND is about 1x10-10m/sec2. For more information
on this fascinating topic, here are some weblinks:
http://nedwww.ipac.caltech.edu/level5/Sept01/Milgrom2/Milgrom_contents.html
http://www.astro.umd.edu/~ssm/mond/litsub.html

[6.c] Interaction Interference Effects in Proper Time Topology.

If the Proper Time topology is a real part of space time, then it will have to
support Schroedinger's wave equation, and the interference effects (see equation 5.9)
that make quantum statistics so different from the usual interference effects.
Since interference effects are so much lower in frequency than the
driving frequencies, we can expect that the interaction interference
will be more easily observable.

Accordingly, let W1 and W2 be two interacting
waves in the Proper Time topology, and let's compute their interference patterns.
If the two waves both represent the same particle type, for instance electrons, then
their ω values will be the same. This is due to the fact that all particles in
the Proper Time topology travel at the same speed. But the two particles can have
different wave vectors, so let their corresponding wave vectors be
k1 and k2. Assuming equal magnitudes
for the waves represented as cosines (and anything else is problematical due to the
fact that this is a very non linear theory that we don't know the details of, but the
average and beat frequencies, which is all we care about at this time, should be correct),
then the interference is:

The (k1+k2)/2 term is of the same
form as the original W waves, while the
(k1-k2)/2 term is an interference
term. Since the wave functions correspond to particles with velocity c, we
can convert k1 and k2
to velocity format using the relation v = ck/k
= c2k/ω. The resulting interference term is:

(6.12) cos((v1 - v2)
ω/2c2 • x).

The above term gives the inteference between two plane waves. For more general waves,
the above is integrated along a suitable path, say by dη, and the general
interference term is therefore:

(6.13) cos(∫12 (v1 -
v2)
ω/2c2 • dη).

Comparing term (6.13) to equation (5.9) shows that interference between waves in Proper
Time topology will be equivalent to interference between waves in Schroedinger's equation
providing:

(6.14) ω/2c2 = m/h.

Consistent with the Proper Time topology, the above equation corresponds to the frequency
of a particle travelling at speed c, but there is now an extra factor of two.
This factor comes from the conversion between quantum and classical interference.
Quantum interference is defined as a straight subtraction, but classical interference
takes the difference and divides by that extra factor of two.

At the present time, the ω frequency is too high to be detected, hence the fact
that a plane wave in quantum mechanics appears to have no spatial or time dependence,
but the beat frequencies are detectable, and are the interference effects seen in
Schroedinger's wave equation.

While it is a true that the the real extension of Schroedinger's equation supports
arbitrary spin wave states, the ones generated in the most obvious manner (i.e. take
equation 5.8 for spin-1, and replace φ with some multiple a of φ, the
new equation will have spin-a), the resulting wave state is not stable. As
ttime goes on, the dissipative term in equation (5.4) will cause the state to evolve
to a standard solution of Schroedinger's equation. But the Proper Time topology has
an extra dimension to play with, so it seems likely that stable solutions with other
than the usual integer angular momentum exist. While the way to go about finding these
is with separation of variables, an obvious spin-1/2 solution exists, and examining it
will promote some familiarity with the equations.

For reference, here are the extended real Schroedinger's equations, where
I've divided out the mass:

Schroedinger's real equation in Proper Time topology is identical, but in four dimensions
instead of three, and the fourth dimension s is to be cyclic. The above equations
show time evolution. For a stable solution, set the partial deriviatives with respect to
t to zero, giving:

Now let's assume that our solution is to describe a single particle moving with
constant velocity. We want to use all the ammunition available from the previous
attack on the 3-dimensional Schroedinger's wave equation, so we need to separate
variables into a 3-dimensional part and the s part.

When looking for a spin 1/2 solution, it's natural to consider setting up
a solution where the magnitude of v matches that of a
known 3-dimensional solutions, but where the average of v over
the s dimension is half that of the 3-dimensional solution. Putting
the s component of v to a constant, as is suitable for
a particle moving at a constant velocity, makes the divergence in equation
(7.2b) give the same result as for the 3-dimensional case. And with the
magnitude of v the same as the 3-dimensional case, it only
remains to set the A to be equal to the 3-dimensional solution to
get all of equation (7.2a) to be the same.

In order to do this, we need to have a vector of unit length, whose direction
depends on s, and whose average is 1/2. For a vector in the z
direction, an obvious candidate is:

(7.3) (cos2(s), sin(s),
sin(s) cos(s), 0)

The above vector has magnitude 1, but averages over s to (1/2, 0, 0, 0).
In addition, it is cyclic in s, is continuous, and can be continuously
rotated around the z-axis to provide a set of vectors with magnitude
1 that average to (cos(φ), sin(φ), 0, 0)/2.
Note that those solutions
to Schroedinger's wave equation in 3-dimensions which are eigenstates of
the z-component of angular momentum, will only have velocities in the x
and y directions, so it is clear that the above vector, along with its
rotations around the z-axis, provide a continuous set of
vectors with the appropriate properties to convert a 3-dimensional solution
to Schroedinger's wave equation to a Proper Time solution.

The z-component of the angular momentum is computed as an integral over all
space. Integrating out the s dependence (which only shows up in the
vectors defined in (7.3a), leaves an average vector in the same direction
as the standard 3-dimensional spin-1 solution, but with half the magnitude.
It is therefore a spin-1/2 solution to the extended Schroedinger's wave
equation in the Proper Time topology.

Relativistic quantum mechanics uses spinors to represent spin 1/2 particles,
and these spinors have two components, with the extra component corresponding
to anti-particles. While I have not shown that other solutions do not exist,
nor that the transformation properties are the same (though the fact that the
metric used in the Proper Time topology is identical to the standard one may
convince some), it is tempting to assume that the anti-particle appears in
the Proper Time topology as the same as the above, but with β defined to
be negative.

As I work out more general solutions for Schroedinger's equation in the Proper
Time topology, I'll add them here. But what we already have is a good start.
Where QED requires different spaces to model scalar and spin-1/2 particles,
the Proper Time topology gets them both as different solutions to the same
equations.

[8] Conclusion and Speculation.

What a long strange trip this has been.

This research began as an attempt to modify QED (Quantum Electrodynamics) so that a single,
not terribly complicated, field would allow the modeling of more than just a single
electron. It soon became apparent that the problem of quantum interference was at the
heart at the difficulty in doing this. An analysis of quantum interference from the
classical point of view, as illustrated by Bohmian mechanics, suggested that hidden
dimensions needed to be included. Bohmian mechanics also provided an interpretation
of the relationship between the extensions of Schroedinger's equation and Schroedinger's
equation itself, and this suggested that the extensions should be used.
An analysis of the results of quantum entanglement experiments, along with the fact of
wave function collapse, suggested that a new interpretation of time would be required,
and that new interpretation would have to be centered around the concept of proper time.

The equations of special relativity suggested that proper time should be treated identical
to the 3 real dimensions, so the topology of Proper Time fell out naturally. When
relativistic dynamics turned out to be simplified in Proper Time, as opposed to Minkowski
space, the selection of Proper Time as the topology was shown to be likely to be correct.
Calculations for self interactions showed no real problems with interpretation, given
the very high energy/small distance nature of the proposed hidden dimension, and the
interpretation of the interaction between multiple particles showed an interference effect
that is identical to that of Schroedinger's equation.

The Schroedinger's wave equation is trivially generalized to the Proper Time topology,
and solutions to the regular Schroedinger's wave equation are still valid. In addition,
a solution with Lz = 1/2 h has been shown to exist.

This is all very heartening, (which is why I'm publishing this now), but there is still
a lot of work left to do.

[8.a] Future Research

The first thing to do is to extend equation (5.4), which defines the dynamics of the
velocity field, to the proper time topology. Since the conversion from
the Proper Time topology to Schroedinger's equation involves the replacement of ω
with m, it seems likely that m will be proportional to the amount of
wave activity in the s dimension. That is, since those things travelling at the
speed of light do not experience the passage of proper time and cannot have a non zero
wave vector in the s direction, their waves will not depend on that dimension.
Waves that do have a non zero wave vector in the s direction will presumably
have mass, and that mass may be extracted in Schroedinger's wave equation as an average
over a very small region of a term that might look, for example, like
∂2/∂s2. In any case, since the theory
assumes that waves travel at the same speed in all directions, including s, it
seems not unlikely that the full wave equation will involve all four dimensions in
the same way. If this is the case, it may be possible to make some deductions as
to the form of that equation. Then take the extended equation and see if we can
get intrinsic spin to work on it. Since Schroedinger's equation cannot support spin 1/2,
(but instead adds it axiomatically) it may be that intrinsic spin is a local feature
of the topology that is averaged out when the s dimension is averaged over.
Then the resulting theory needs to be shown that it, like classical mechanics on the
Proper Time topology, is also compatible with special relativity.

One of the problems associated with extending quantum mechanics to extra dimensions is
that one ends up with more particles, generally at the higher energies, than one desires.
An example of this kind of issue will be put in this paper as Appendix B. But if the
objective is to create a fundamental theory, then it is necessary that these higher
energy solutions correspond to real particles observed in the world. The natural place
to look for higher energy solutions would be the the higher energy correspondents to the
electron. That is, given a field equation that supports solutions that represent electrons,
it would be best if the same field equation also supports higher energy solutions that
represent the μ and τ as well. Ideally, an accurate field equation should give
the relative (bare) masses of these particles.

The relationship between the Kahluza-Klein extension of special relativity to include
electromagnetism, and the Proper Time topology needs to be explored. Maybe this is an
indication that more hidden dimensions need to be added to the Proper Time topology,
or maybe the one that is already there is equivalent to that posited in Kahluza-Klein.
The Proper Time topology is symmetric with respect to parity (i.e. space inversion),
but it was shown in section [5.c] that the Proper Time topology would nevertheless
support fermion wave functions. Time reversal, on the other hand, will be explicitly
broken by any theory that connects wave functions, through time, to their collapsed
predecessors. The operation of charge conjugation awaits the inclusion of electromagnetism
to the Proper Time topology.

Rather than risk associating the teachers and institutions of higher learning that have
had an influence on me with what may be treated as another crack-pot theory, I will here
only acknowledge that the physics community has treated me and this theory with nothing
other than respect and leave it at that.
Carl Brannen

[A] Appendix A: Special Relativity compared to Proper Time topology

For those who have difficulty seeing how it is that the Proper Time topology,
which explicitly possesses a preferred frame of reference, could be equivalent to
special relativity, which makes the assumption that no such frame exists, this
appendix provides detailed calculations for time dilation and length contraction
both in special relativity and in Proper Time topology.

[A.a] Time dilation
Problem: A spaceship travels 3 light years away form earth, at a speed of 0.3c,
and then returns at the same speed. What is the proper time experienced on the Earth during
the voyage, and what is the proper time experienced on the spaceship?

Special Relativity Solution: The voyage requires 3/0.6 = 5 years each way for a total of 10 years.
This is the proper time experienced on the Earth. The spaceship experiences a time dilation of
√(1 – 0.62) = 0.8, so the proper time experienced on the spaceship is
10 x 0.8 = 8 years.

Proper Time Solution: The spaceship starts at the point (x, y, z, s) = (0, 0, 0, 0).
Align the x axis with the direction of travel. The velocity of the spaceship on the
outgoing voyage is therefore given by the vector (0.6, 0, 0, 0.8). The 0.8 value is required
to make the speed of the spaceship work out in total be 1. The spaceship’s position as a
function of the global time t is therefore:

(0, 0, 0, 0) + (0.6, 0, 0, 0.8) t1

Setting this equal to (3, 0, 0, s1) gives t1, the global
coordinate time for the arrival of the spaceship at its destination, and t1
is therefore 5 years. Note that the value of s1 is unspecified, as the
total length of the hidden dimension is negligible as compared to the many light years of
travel.
Since the proper time component of the velocity of the spaceship is
0.8, the total elapsed proper time on the outgoing voyage of the spaceship is therefore
0.8 x 5 = 4 years. Similarly, the return trip uses a velocity of (-0.6, 0, 0, 0.8) and
results in a coordinate time passage of 5 years and a proper time for the spaceship of another
4 years. The result is, of course, identical to the Special Relativity result.

[A.a] Lorentz Contraction
A rod flies lengthwise through a laboratory with a speed of 12/13c. The lab
measures the length of the rod as 6 meters. How long is the rod measured in a coordinate
system moving with the rod?

Since the Proper Time topology does have a preferred coordinate system, the question is not
as clear as it is in special relativity. But in any given coordinate system, the constancy
of the speed of light provides a technique for measuring length. Accordingly, the rod can
be measured in its own frame of reference by calculating the time required for light to
travel the length of the rod. Since proper time is a property of individual particles,
rather than dimensional objects such as rods, the length of the rod will have to be measured
by computing the time required for the light to travel down the rod, be reflected at the
end, and then travel back to the point of origin on the rod. The proper time experienced
by the end point of the rod during this flight will indicate (when multiplied by c = 1)
twice the length of the rod.

So let the rod begin at position (0, 0, 0, 0) through (6m, 0, 0, 0), and set the velocity vector for
the rod to be (12/13, 0, 0, 5/13) so that it moves in the +x direction. The light
signal starts at (0, 0, 0, 0) and proceeds with a velocity vector of (1, 0, 0, 0) until it
meets with the other end of the bar at time t1. The light direction is then
reversed, and it travels with velocity (-1, 0, 0, 0) until it meets up with the trailing end
of the bar at time t2. The length of the bar, in the reference frame of the
bar, is then 1/2 the proper time experienced by the trailing end of the bar from 0 to
t2. The equations for t1 and t2 are therefore:

Since our real world does not distinguish between the hidden “proper time” coordinate, the
equalities need only be established for the first three coordinates.

The solution is t1 = 13 x 6 meters, and t2 = 2028/25 meters.
The proper time experienced on the trailing edge of the rod is, by time dilation, 5/13 of
t2, which gives 156/5. Half of this is the proper length of the bar, which is
the same as the value given by special relativity. Therefore, both theories show the Lorentz
contraction of the bar to be the same.

[B] Appendix B: A (bad) extension of Schroedinger's equation

This modification of Schroedinger's equation removes the imaginary numbers using a hidden
dimension. Note that this is not the hidden dimension of the Proper Time topology, but
this appendix is intended to illustrate what kind of effects hidden dimensions have on
Schroedinger's equation. For reference, here's Schroedinger's wave equation:

(B.1) i h ∂Ψ / ∂ t =
(- h / 2m Δ + V(r)) Ψ
(r,t) .

When we add an extra dimension to the above equation, the old functions Ψ will
have to be extended to new functions Ψ' which instead of being functions of
time and 3 dimensions, will now have to be functions of time and 4 dimensions. Call the
coordinate for the new dimension ζ, and suppose that it is cyclic with length 2π.
That is, Ψ'(x, ζ, t) =
Ψ'(x, ζ + 2π, t). The simplest guess for how
Ψ' depends on ζ is a sine or cosine.

The wonderful thing about sin(ζ) and cos(ζ) is that they convert into
one another when differentiated with respect to ζ, but that going one way there is
an extra minus sign. This is identical to how imaginary numbers work, if one interprets
1 as sin, i as cos, and replaces "multiplication by i" with
a derivative with respect to ζ. Here's a table showing the equivalence.

.

Ψ

Ψ'

Values:

z = x + iy

x sin(ζ) + y cos(ζ)

Action:

i

∂/∂ζ

"Values" in the above table are the values of the Ψ
and Ψ' functions, while "Action" refer to the operators in their respective
equations, that is, in Schroedinger's wave equation and in this extension of Schroedinger's
wave equation. The above table shows how to convert Schroedinger's wave equation from
a complex equation in three dimensions (plus time), into a real equation in four dimensions
plus time. The converted equation is as follows, note that the operator i has been
replaced with a derivative with respect to ζ:

(B.2) h ∂2Ψ / ∂ζ ∂ t =
(- h / 2m Δ + V(r)) Ψ
(r,t) .

Any (complex valued) solution of Schroedinger's wave equation (B.1) can be converted into a
(real valued) solution of the extended equation (B.2). The conversion, according to the above
table, is as follows:

(B.3) Ψ'(x, ζ, t) =
Re(Ψ(x, t)) sin(ζ)
+ Im(Ψ(x, t)) cos(ζ).

While every solution of Schroedinger's equation gives a solution to the extended equation,
the reverse is not the case. That is, if Ψ' has ζ dependency other than
just sine or cosine, the table above does not give a translation into a solution of Schroedinger's
wave equation, at least the original Schroedinger's equation.

Instead, the extra solutions, when translated back from ζ into complex functions, give
solutions to altered versions of Schroedinger's wave equation. The extra solutions correspond
to wave equations for particles with masses or potentials different from the original
solutions. It's also possible to consider translations of actions other than just i.
For example, the action "multiplication by -1" can be obtained by two consecutive mutliplications
by i, and is therefore translated as ∂2/∂ζ2.
A more general table, this suggests that the above table be extended to show some of these
more general translations:

.

Ψ

Ψ'

Values:

z = x + iy

x sin(n ζ) + y cos(n ζ)

Action:

n i

∂/∂ζ

...

-n2

∂2/∂ζ2

...

-n3i

∂3/∂ζ3

...

n4

∂4/∂ζ4

...

...

...

Another way of seeing the above translations is to use separation of variables on equation (B.2)
in the usual way as is common in any textbook showing how to find general solutions of Schroedinger's
equation.

Depending on how you choose to extend Schroedinger's wave equation, you can create equations
that have, in addition to the usual solutions, solutions that correspond to particles with
higher mass or different potentials (and therefore different charge). But while these are
interesting, the solutions do not give the things that we'd like to have, such as the mass
of the muon. For that, we need to have the non perturbational theory. But the fact that
dimensional extensions of Schroedinger's wave equation creates equations with solutions for
more than just one mass particle is heartening. This, along with the result that the Proper
Time topology supports quantum interference effects, gives an outline for a multiparticle
wave function that emulates the spin statistics used so successfully in quantum field theory.